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7
votes
0answers
77 views

A tensor product for triangulated categories?

Many triangulated categories which show up in mathematics, such as derived categories of various sorts, arise as the homotopy category of a stable $\infty$-category. Stable $\infty$-categories give ...
4
votes
1answer
135 views

Stabilization of a generic pointed model category

Let $\mathcal C$ be a pointed model category. It is well-known that its homotopy category $\mathrm{Ho}(\mathcal C)$ is naturally a $\mathrm{Ho}(\underline{\mathrm{sSet}}_*)$-category, where ...
3
votes
1answer
394 views

$\pi_8(S^5)=\pi_8(SO(6))=\mathbb{Z}/24$?

I am currently thinking about a physics model related to framed bordism $\Omega_3^{fr}=Z/24=\pi^s_3$, and the first stable example is $\pi_8(S^5)$, so I was curious about the generator, and happened ...
5
votes
0answers
146 views

Moore spectra are not E-infinity (oldest known proof)

Fix a prime $p$. Let $M_p(i)$, the $i$-th Moore spectrum at the prime $p$, be the cofiber of the map $$ S^0 \overset{p^i}\longrightarrow S^0 $$ where $S^0$ be the sphere spectrum. In the Mathoverflow ...
6
votes
3answers
710 views

Ambidexterity and Quantization

Here the nlab says about Hopkins-Lurie's ambidexterity paper: "The discussion in the article is apparently motivated as part of what it takes to make precise the discussion of quantization in ...
5
votes
1answer
211 views

Naive G-spectrum representing geometric equivariant cobordism

Let $G$ be a finite group. By the transversality results of Wasserman $G$-equivariant bordism (say real) should be a naive homology theory, and as such it should be represented by a naive G-spectrum. ...
-2
votes
1answer
73 views

stable splitting into a wedge sum [closed]

Suppose $X$ is a CW-complex such that there is a stable splitting of $X$ into wedge sum $$ \Sigma^t X\cong \bigvee _{k=1}^\infty Y_k. $$ (1). Does this imply $$ X\to \Sigma^tX\to \bigvee ...
7
votes
0answers
187 views

May's infinite loop machine for Friedlander's result for Adams conjecture

Eric M. Friedlander in the paper The infinite loop Adams conjecture via Classification Theorem for $\Gamma$-spaces proved the infinite loop Adams conjecture using techniques involved $\Gamma$-space. ...
3
votes
0answers
88 views

Maps between equivariant loop spaces

I have an elementary question about equivariant loop spaces that I feel it should be well known. Given a finite group $G$ and a finite $G$-set $J$ let $S^J=\mathbb{R}[J]^+$ be the permutation ...
10
votes
1answer
309 views

Cohomology of the Image of J spectrum

Let $J$ denote the image of $J$-homomorphism spectrum and let $j$ denote its connective cover. I am interested in knowing the cohomology of $j$ i.e. $$ [j, HZ/p]_*$$ as a module over Steenrod algebra. ...
2
votes
3answers
338 views

Integral transform on noncommutative spaces

In their paper "Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry" the authors show that for perfect stacks $X$ and $Y$ over $k$, and their $k$-linear $\infty$-categories of ...
20
votes
0answers
322 views

Nilpotence of the stable Hopf map via framed cobordism

The Pontryagin-Thom construction shows that the stable homotopy groups of spheres are the same as the groups of stably framed manifolds up to cobordism. Specifically the Hopf map corresponds to the ...
2
votes
0answers
238 views

Functor of points of a tensor triangulated category

Is there is a functor of points approach to tensor triangulated categories parallel to Balmer's theory of prime spectra? Given a tensor triangulated category $\mathcal{T}$ an $R$-point can be ...
5
votes
2answers
227 views

homology of a mapping spectrum

If $X$ and $Y$ are two spectra, I denote by $F(X,Y)$ their mapping spectrum. This is uniquely determined by the existence of a natural isomorphism $[X\wedge Y, Z]\cong [X,F(Y,Z)]$. I denote by $H_*$ ...
5
votes
0answers
100 views

Spectral Sequences of Parametrized Spectra

I apologize if this question is of the form "what are some interesting problems in bla" but I was wondering if anybody have studied the following set-up: Suppose that I have a parametrized spectra ...
5
votes
1answer
196 views

When does a map in the stable homotopy group gets killed when smashed with cone of itself?

Consider an element $e \in \pi_n^s(S^0)$, in the stable homotopy groups of sphere. Let $C$ denote the spectrum which is cone of $e$, i.e. $C$ fits in the cofiber sequence $$ S^n \to S^0 \to C.$$ ...
7
votes
1answer
473 views

Homologically distinct infinite loop structures on a space

Let $X$ be a connected pointed topological space equipped with two different actions of $E_\infty$-operad. Each action provides a collection of deloopings $X_i$, where $X_0 = X$ and $\Omega X_i$ is ...
13
votes
1answer
281 views

Anything between vector bundles and sphere bundles?

There are two extremities: on the "easy end" one has vector bundles which are classified by maps to the (more or less) well understood spaces like Grassmanians; on the "hard end" there are spherical ...
3
votes
0answers
148 views

Twisting of the power functor

Let $k$ be a field of characteristic $p$ and $D^b(k)$ be the infinity (equivalently, DG) category of perfect complexes over $k$. Let $C_p(=\mathbb{Z}/p)$ be the cyclic group on $p$ elements. For a ...
-1
votes
1answer
349 views

$E_n$ structures on Symmetric Monoidal Stable infinity-categories [closed]

Let $C$ be a stable $∞$-category with a monoidal structure on it, hence a monoidal stable $∞$-category; if this monoidal structure is "maximally symmetric" (in Wikipedia's sense) $C$ is a symmetric ...
5
votes
1answer
344 views

Uniqueness of Complex Orientation of Morava K-theory

It is known that the $n^{\text{th}}$ Morava $K$-theory at a prime $p$, denoted $K(n)$, is complex oriented. In other words, it admits a theory of Chern classes, or equivalently a morphism of homotopy ...
5
votes
1answer
196 views

What does an endomorphism in a triangulated category give rise to?

Let $D\xrightarrow[]\varphi D\xrightarrow[]kE\xrightarrow[]j\Sigma D$ be an exact triangle in a triangulated category. I am trying to figure out what structure emerges from this on the base of the ...
6
votes
1answer
381 views

K-theory of the h-cobordism category

I was reading through Kervaire and Milnor's "Groups of Homotopy Spheres", in which the authors begin to compute the groups $\Theta_n$ of h-cobordism classes of homotopy $n$-spheres (with group ...
15
votes
0answers
283 views

Chromatic Spectra and Cobordism

I apologize in advance, if some of the things I've written are incorrect. The cobordism hypothesis states that $\mathbf{Bord}^\mathrm{fr}_n$ is the free symmetric monoidal $(\infty,n)$-category with ...
33
votes
2answers
909 views

What is an infinite prime in algebraic topology?

The links between algebraic topology (stable homotopy theory in particular) and number theory are nowadays abundant and fruitful. In one direction, there is chromatic homotopy theory, exploiting the ...
2
votes
1answer
180 views

About embedding pure motives into the triangulated category of mixed motives and some further questions about motivic cohomology

I have tried reading some texts about motives, mainly motivic cohomology (Bloch's " Lectures on Algebraic Cycles", Voevodsky's "Motivic Cohomology" etc). However some things confused me. I don't know ...
8
votes
1answer
236 views

Power operations and Lambda-structure-like lifts of Frobenius in $E_\infty$-geometry?

A $\Lambda$-structure on a commutative ring $R$ is a ring endomorphism wich restricts to the $p$-Frobenius homomorphism after localizing at $(p)$. One may think of this as a "flow" $\Phi \colon ...
4
votes
1answer
168 views

AdicCompletion$\dashv$Torsion adjunction on spectra?

It seems to me that in slight paraphrase the central result of the article Marco Porta, Liran Shaul, Amnon Yekutieli, On the Homology of Completion and Torsion (arXiv:1010.4386) (theorems 6.11 and ...
17
votes
4answers
589 views

Fibrations and Cofibrations of spectra are “the same”

My question refers to a folklore statement that I have now seen a couple of times, but never really precise. One avatar is: "For spectra every cofibration is equivalent to a fibration" (e.g. in the ...
5
votes
0answers
119 views

Parametrized cancelations in stable Morse theory

Let $B$ be a closed manifold. Let $\pi : M\to B$ be a submersion such that each fiber is a manifold without boundary. Let $f : M \to \mathbb{R}$ be a function such that the restrictions $f_x$ to each ...
6
votes
1answer
341 views

Detection of stable homotopy by K-theory spectra

This is primarily a reference request. Does anyone know of any writing about algebraic K-theory spectra picking up elements in the stable homotopy groups of spheres in their Hurewicz image coming from ...
21
votes
1answer
746 views

From the perspective of bordism categories, where does the ring structure on Thom spectra come from?

To fix ideas, let's consider the Thom spectrum of framed bordism $M$, the spectrum whose homotopy groups are the framed bordism groups. $M$ has a ring spectrum structure inducing the product of ...
5
votes
1answer
265 views

endomorphisms of modules over symmetric ring spectra

I have a probably very basic question about modules over symmetric ring spectra: Let $R$ be a commutative symmetric ring spectrum and let $M$ and $N$ be module spectra over $R$. Moreover, let ...
10
votes
1answer
272 views

Is there a non-zero ghost map between finite suspension spectra?

A morphism $f\colon X\to Y$ of spectra such that for every integer $n$ the induced map $\pi_n(f)\colon\pi_n(X)\to\pi_n(Y)$ on stable homotopy groups is zero is called a ghost map. Not every ghost map ...
9
votes
1answer
284 views

Verdier localization for stable $\infty$-categories

Verdier localization is one of the more intuitive ways to localize a triangulated category, "killing" a suitable class of objects via a functor which is universal with respect to this property. I ...
5
votes
1answer
425 views

Dennis trace map K----> THH

I have some questions about Dennis trace map in algebraic K-Theory. I was wondering if there is some conceptual way to look at this map $K(-)\rightarrow THH(-)$ (natural transformation from K-Theory ...
24
votes
4answers
1k views

What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?

I know of two very general frameworks for describing generalizations of what a "cohomology theory" should be: Grothendieck's "six functors", and the theory of spectra. In the former, one assigns to ...
10
votes
1answer
421 views

The homotopy of universal Thom spectrum

Let $S^0_p$ be the $p$-adic sphere spectrum. Let $GL_1(S^0_p)$ be the set of unit componen of $\Omega^{\infty}S^0_p$. For any map $ X \to BGL_1(S_p^0)$ we get a Thom spectrum call it $Mf$. Now ...
5
votes
0answers
195 views

Which ring spectra have some kind of exponential map turning addition into multiplication?

This accepted answer to the question about $BU_\otimes$ made me recall that I want to read about this general phenomenon for a long time. What will follow is sort of vernacular but whether it can be ...
2
votes
1answer
173 views

Is there a geometric interpretation of Johnson-Wilson E(n) analogous to vector bundles for K-theory?

I am reading Ravenel's Localization with Respect to Certain Periodic Homology Theories where he states; For $n\ge2$, the spectra E(n) represent periodic homology theories which at present have ...
2
votes
1answer
245 views

Generator of $\pi_3(SU(4))$ in Mimura-Toda

In this paper of Mimura and Toda, tables are given for low-dimensional homotopy groups of $SU(3)$, $SU(4)$ and $Sp(2)$. As far as I understand it, Theorem 6.1 gives the generator of $\pi_3(SU(4))$ as ...
4
votes
3answers
312 views

Adams Spectral Sequence for Triangulated Categories

We have the Adams SS with $$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$ where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients. I was wondering if there is a SS for ...
19
votes
0answers
605 views

Calabi-Yau cohomology?

My question here is going to be this -- but I'll give a bit of background to explain myself in a moment: What has been done/what results are available on Calabi-Yau cohomology in degree $n \geq 3$ ...
3
votes
1answer
208 views

Stable homotopy of classifying space for nilpotent groups

Let $BG$ denote the classifying space of a (discrete) group and $BG_+$ its disjoint union with a point. Question: What is known about the stable homotopy groups $\pi^S_*(BG_+)$ ? If $G$ is finite ...
6
votes
1answer
262 views

When was the word “stable” first used to describe stable homotopy theory?

The word "stable" has many uses in mathematics, but in the context of stable homotopy theory, one might take it to mean one of two things: Homotopy groups stabilize after taking suspensions ...
4
votes
2answers
154 views

Homology exponents for $QX$

We say that a space $X$ has a homology $p$-exponent if some power of $p$ annihilates the $p$-torsion in $H_\ast(X;\mathbb{Z})$. I am interested in the homology exponents of the free infinite loop ...
10
votes
0answers
190 views

“topological” Ochanine genus?

The Witten genus has famously been lifted to the string orientation of tmf ("topological Witten genus"). For the Ochanine genus I am aware of a lift to a "spin orientation of Tate K-theory", namely to ...
12
votes
2answers
403 views

Truncations of E_infinity algebras

In section 4.1 of Lurie's DAG VIII, he implies the existence of an $E_\infty$-ring spectrum $A$ such that the coconnective truncation $\tau_{\leq 0} (A)$ does not admit the structure of an ...
21
votes
1answer
889 views

What is to tmf as KR is to KO?

The $E_\infty$-ring spectrum $KU$ of complex K-theory carries a canonical involution induced from complex conjugation of complex vector bundles. The homotopy fixed points of this $\mathbb{Z}_2$-action ...
2
votes
0answers
197 views

Units of a ring spectrum

Is there a good notion of the spectrum of units $R^\ast$ in a (possibly non-connective) $E_\infty$-ring spectrum $R$? A standard definition (see section 1.2 in http://arxiv.org/abs/0810.4535) seems ...